Transversals of subtree hypergraphs and the source location problem in digraphs

A hypergraph H = (V,E) is a subtree hypergraph if there is a tree T on V such that each hyperedge of E induces a subtree of T . To find a minimum size transversal for a subtree hypergraph is, in general, NP-hard. In this paper, we show that if it is possible to decide if a set of vertices W ⊆ V is a transversal in time S(n) ( where n = |V | ), then it is possible to find a minimum size transversal in O(n S(n)). This result provides a polynomial algorithm for the Source Location Problem : a set of (k, l)-sources for a digraph D = (V,A) is a subset K of V such that for any v ∈ V \K there are k arc-disjoint paths that each join a vertex of K to v and l arc-disjoint paths that each join v to K. The Source Location Problem is to find a minimum size set of (k, l)-sources. We show that this is a case of finding a transversal of a subtree hypergraph, and that in this case S(n) is polynomial.